Navigational ambiguity in robot path-finding: Complement metric dimension of symmetric convex planar spaces

In graph-based models of robot navigation, landmark selection plays a critical role in enabling accurate movement across a spatial environment. A metric generator ensures unique positional identification, aiding effective path-finding. Conversely, a metric resolving complement consists of those points that fail to resolve all other positions, thus representing sources of navigational ambiguity or distraction. The cardinality of the largest such set defines the complement metric dimension of the space. In this paper, we study this concept in the context of six structured symmetric convex planar (SCP) spaces, each representing a navigable environment for a robot. We explicitly construct maximum metric resolving complements and determine exact values of the complement metric dimension for each SCP space. Our results offer theoretical insights into the identification of worst-case landmark configurations in robotic path-finding systems.